Mathematical Notations

Notation

Definition

\(n\) or \(m\)

The number of observations in a dataset. Typically \(n\) is used, but sometimes \(m\) is required to distinguish two datasets, e.g., the training set and the inference set.

\(p\) or \(r\)

The number of features in a dataset. Typically \(p\) is used, but sometimes \(r\) is required to distinguish two datasets.

\(a \times b\)

The dimensionality of a matrix (dataset) has \(a\) rows (observations) and \(b\) columns (features).

\(|A|\)

Depending on the context may be interpreted as follows:

  • If \(A\) is a set, this denotes its cardinality, i.e., the number of elements in the set \(A\).

  • If \(A\) is a real number, this denotes an absolute value of \(A\).

\(\|x\|\)

The \(L_2\)-norm of a vector \(x \in \mathbb{R}^d\),

\[\|x\| = \sqrt{ x_1^2 + x_2^2 + \dots + x_d^2 }.\]

\(\mathrm{sgn}(x)\)

Sign function for \(x \in \mathbb{R}\),

\[\begin{split}\mathrm{sgn}(x)=\begin{cases} -1, x < 0,\\ 0, x = 0,\\ 1, x > 0. \end{cases}\end{split}\]

\(x_i\)

In the description of an algorithm, this typically denotes the \(i\)-th feature vector in the training set.

\(x'_i\)

In the description of an algorithm, this typically denotes the \(i\)-th feature vector in the inference set.

\(y_i\)

In the description of an algorithm, this typically denotes the \(i\)-th response in the training set.

\(y'_i\)

In the description of an algorithm, this typically denotes the \(i\)-th response that needs to be predicted by the inference algorithm given the feature vector \(x'_i\) from the inference set.
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